
## 3. Geometric vs. Arithmetic representations

Take $f(z)\in\mathbb{Q}[z],$ assume $0$ is not periodic. Consider two groups \begin{align} G &:= \lim_{n\rightarrow\infty} \text{Gal}(f^n)\\ G’&:= \lim \text{Gal}(f^n(z)-t). \end{align}
1. #### Problem 3.1.

[Tom Tucker] Is there any example where $G$ does not have finite index in $G'?$
• #### Problem 3.2.

[Tom Tucker] Understand the relation with PCFness. Note this is related to Problem 1.1. Perhaps start with $f(z)=z^2-1.$

Cite this as: AimPL: The Galois theory of orbits in arithmetic dynamics, available at http://aimpl.org/galarithdyn.